Unsteady Magnetohydrodynamic Convective Boundary Layer ... Unsteady Magnetohydrodynamic Convective

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  • Unsteady MagnetohydrodynamicConvective Boundary Layer Flow past aSphere in Viscous and Micropolar Fluids

    Nurul Farahain Mohammad

    Department of Mathematical Sciences, Faculty of Science,Universiti Teknologi Malaysia

    Name of supervisors:Assoc. Prof. Dr. Sharidan Shafie and Dr. Anati Ali.

    1

  • Outline of Presentation

    Introduction

    Problem 1 / Chapter 4

    Problem 2 / Chapter 5

    Problem 3 / Chapter 6

    Problem 4 / Chapter 7

    Problem 5 / Chapter 8

    Conclusion

    Suggestions for Future Works

    Publication / Awards / Attended Conferences

    2

  • Introduction

    Objectives

    to examine the effects of MHD on unsteady boundary layerflow over a sphere.

    to analyze the behaviour of viscous fluid and micropolarfluid under the influence of MHD.

    to investigate the interaction between MHD flow and heattransfer with or without buoyancy force.

    3

  • Introduction

    Scope

    electrically-conducting viscous and micropolar fluids incompressible unsteady 2D laminar boundary layer flow

    past a sphere uniform magnetic field, transverse of the fluid flow induced magnetic field is neglected no polarized or applied voltage enforced on the fluid flow numerical solution (Keller-Box method) no real experiments conducted to validate the numerical

    results

    4

  • Research Methodology

    Mathematical Analysis

    dimensionless variables stream function similarity transformation

    Keller-Box Method

    finite difference method Newtons method block-tridiagonal factorization scheme

    5

  • Problems Solved

    O

    a

    U

    x/ayr(x)

    x

    Tw

    T

    T

    g

    Figure : Physical Coordinate

    Viscous fluid Micropolar fluidsBoundary Layer Flow Prob. 1 / Chap. 4 Prob. 4 / Chap. 7Forced Convection Prob. 2 / Chap. 5 -Mixed Convection Prob. 3 / Chap. 6 Prob. 5 / Chap. 8

    6

  • Problem 1 / Chapter 4

    (r (x) u

    )x

    +(r (x) v

    )y

    = 0, (1)

    ut

    + uux

    + vuy

    = 1

    px

    +

    (2ux2

    +2uy2

    ) B0

    2

    u, (2)

    vt

    + uvx

    + vvy

    = 1

    py

    +

    (2vx2

    +2vy2

    ) B0

    2

    v , (3)

    subject to the following initial and boundary conditions:

    t < 0 : u = v = 0, for any x , y ,

    t 0 : u = v = 0, at y = 0,u = ue(x), as y .

    (4)

    7

  • Dimensional Governing Equations

    Non-dimensional Governing Equations

    Governing Equations in Stream Function

    Non-similar Governing Equations

    Dimensionless variables

    Stream function

    Similarity variables

    8

  • Discretized Governing Equations

    Linearized Numerical Scheme

    Block Tridiagonal Factorization Scheme

    Equations solved

    ts, xs, f ,Cf Re1/2,h, s,NuRe1/2

    Finite Difference Method

    Newtons Method

    LU factorization

    Block Elimination Method

    Analyse

    9

  • Problem 1 / Chapter 4

    Table : The separation times of flow past the surface of a sphere.

    x M = 0 M = 0 M = 0.1 M = 0.5 M = 1.0 M = 1.3(Ali, 2010) (present)

    180 0.3966 0.3960 0.4161 0.5241 0.7963 1.2470171 0.4016 0.4010 0.4217 0.5331 0.8186 1.3103162 0.4177 0.4170 0.4394 0.5623 0.8940 1.5677153 0.4471 0.4463 0.4721 0.6178 1.0627 -144 0.4947 0.4937 0.5257 0.7152 1.4428 -135 0.5709 0.5694 0.6128 0.8937 - -126 0.6987 0.6960 0.7632 1.2953 - -117 0.9442 0.9372 1.0779 - - -108 - 1.6751 - - - -

    Ali, A. (2010). Unsteady micropolar boundary layer flow and convective heattransfer. Universiti Teknologi Malaysia, Faculty of Science: PhD Thesis.

    10

  • Problem 1 / Chapter 4

    0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

    0.1

    0.2

    0.3

    0.4

    0.5

    0.6

    0.7

    0.8

    0.9

    1

    f

    M = 0, 0.1, 0.5, 1.0, 1.5

    0 10 20 30 40 50 60 70 80 900.6

    0.4

    0.2

    0

    0.2

    0.4

    0.6

    0.8

    1

    f

    M = 0, 0.1, 0.5, 1.0, 1.5

    x = 0 x = 180

    11

  • Problem 1 / Chapter 4

    0 20 40 60 80 100 120 140 160 1801.5

    1

    0.5

    0

    0.5

    1

    1.5

    2

    2.5

    3

    x

    Cf R

    e1/2

    t = 0.1, 0.5, 1.0, 1.5, 2.0

    0 20 40 60 80 100 120 140 160 1800

    0.5

    1

    1.5

    2

    2.5

    3

    3.5

    x

    Cf R

    e1/2

    = 0.1, 0.5, 1.0, 1.5, 2.0 t

    Without MHD MHD

    12

  • Problem 2 / Chapter 5

    (r (x) u

    )x

    +(r (x) v

    )y

    = 0, (5)

    ut

    + uux

    + vuy

    = 1

    px

    +

    (2ux2

    +2uy2

    ) B0

    2

    u, (6)

    vt

    + uvx

    + vvy

    = 1

    py

    +

    (2vx2

    +2vy2

    ) B0

    2

    v , (7)

    Cp(Tt

    + uTx

    + vTy

    )= c

    (2Tx2

    +2Ty2

    ), (8)

    subject to the following initial and boundary conditions:

    t < 0 : u = v = 0,T = T for any x , y ,

    t 0 : u = v = 0,T = Tw at y = 0,

    u = ue(x),T = T as y .

    (9)

    13

  • Problem 2 / Chapter 5

    Ms NuRe1/2

    xCf(x = 0) (x = 180) (x = 0) (x = 180)

    Pr s NuRe1/2

    t NuRe1/2

    14

  • Problem 3 / Chapter 6 (r u)x

    + (r v)y

    = 0, (10)

    (ut

    + uux

    + vuy

    )= p

    x+

    (2ux2

    +2uy2

    ) B20u g(T T) sin

    (xa

    ),

    (11)

    (vt

    + uvx

    + vvy

    )= p

    y+

    (2vx2

    +2vy2

    ) B20v + g(T T) cos

    (xa

    ),

    (12)

    Cp(Tt

    + uTx

    + vTy

    )= c

    (2Tx2

    +2Ty2

    ), (13)

    subject to the following initial and boundary conditions:

    t < 0 : u = v = 0,T = T for any x , y ,

    t 0 : u = v = 0,T = Tw at y = 0,

    u = ue(x),T = T as y .

    (14)

    15

  • Problem 3 / Chapter 6

    M ts xs Cf Re1/2NuRe1/2

    (x = 0) (x = 180)

    ts xs Cf Re1/2NuRe1/2

    (x = 0) (x = 180)

    16

  • Problem 4 / Chapter 7(r (x) u

    )x

    +(r (x) v

    )y

    = 0, (15)

    (ut

    + uux

    + vuy

    )= p

    x+(+)

    (2ux2

    +2uy2

    )+

    NyB20u, (16)

    (vt

    + uvx

    + vvy

    )= p

    y+(+)

    (2vx2

    +2vy2

    )N

    xB20v , (17)

    j(Nt

    + uNx

    + vNy

    )=

    (2Nx2

    +2Ny2

    )

    (2N +

    uy vx

    ), (18)

    subject to the following initial and boundary conditions:

    t < 0 : u = v = N = 0, for any x , y ,

    t 0 : u = v = 0,N = nuy, at y = 0,

    u = ue(x),N = 0, as y .

    (19)

    17

  • Problem 4 / Chapter 7

    M ts xs f Cf Re1/2h

    (x = 0) (x = 180)

    ( = 0) ( = 0, = ) ( ) ( )

    K ts f h

    18

  • Problem 5 / Chapter 8(r (x) u

    )x

    +(r (x) v

    )y

    = 0, (20)

    (ut

    + uux

    + vuy

    )= p

    x+ (+ )

    (2ux2

    +2uy2

    )+

    Ny B20u + g

    (T T

    )sin x ,

    (21)

    (vt

    + uvx

    + vvy

    )= p

    y+ (+ )

    (2vx2

    +2vy2

    ) N

    x B20v g

    (T T

    )cos x ,

    (22)

    j(Nt

    + uNx

    + vNy

    )=

    (2Nx2

    +2Ny2

    )

    (2N +

    uy vx

    ), (23)

    Cp(Tt

    + uTx

    + vTy

    )= c

    (2Tx2

    +2Ty2

    ), (24)

    19

  • Problem 5 / Chapter 8

    subject to the following initial and boundary conditions:

    t < 0 : u = v = N = 0,T = T for any x , y ,

    t 0 : u = v = 0,N = nuy,T = Tw at y = 0,

    u = ue(x),N = 0,T = T as y .

    (25)

    20

  • Problem 5 / Chapter 8

    M ts xs Cf Re1/2NuRe1/2

    (x = 0) (x = 180)

    (Pr = 0.7) (Pr = 7)

    K = 1,n = 0 = 1 = 1Pr = 0.7 M = 0.9 M = 2Pr = 7 M = 1.2 M = 1.5

    K = 1,n = 0.5 = 1 = 1Pr = 0.7 M = 0.9 M = 2Pr = 7 M = 1.2 M = 1.7

    21

  • Conclusion 5 different unsteady MHD models in viscous and

    micropolar fluids over a sphere. 3-dimensional numerical schemes. MATLAB programmings. Analyses of results obtained in MATLAB. Results compared with published work are in good

    agreement. MHD is able to resolve issue involving separation of flow. Given appropriate magnetic strength, separation of flow is

    no longer detected. MHD has potential to increase heat transfer at the surface

    of sphere. Opposing flow requires stronger magnetic field to

    encounter separation of flow to be compared to assistingflow.

    22

  • Conclusion

    M ts xs f Cf Re1/2

    s NuRe1/2

    (x = 0) (x = 180) (x = 0) (x = 180)

    M ts xs f Cf Re1/2

    h NuRe1/2

    (x = 0) (x = 180) (x = 0) (x = 180)

    ( = 0) ( = 0, = ) (Pr = 0.7)

    ( ) ( ) (Pr = 7)

    = 1 = 1

    Pr = 0.7 lowest M highest M

    Pr = 7 2nd lowest M 2nd highest M

    23

  • Suggestions for Future Works

    induced magnetic field. electric field. method to determine appropriate values of M. other geometries: blunt bodies, circular cylinder, elliptic

    cylinder. other effects: Hall effect, internal heat generation,

    Newtonian heating, heat flux, and much more.

    24

  • Publication / AwardsISI Indexed Publication : N.F. Mohammad, A.R.M. Kasim, A. Ali, and S. Shafie. (2014). Separation times analysis of

    unsteady magnetohydrodynamics mixed convective flow past a sphere. AIP ConferenceProceedings 1605: 349-354.

    N.F. Mohammad, M. Jamaludin, A. Ali, and S. Shafie. (2012). Separation Times Analysis ofUnsteady Boundary Layer Flow Past an Elliptic Cylinder Near Rear Stagnation Point. WorldApplied Sciences Journal 17 (Special Issue of Applied Math): 27-32.

    N.F. Mohammad, A.R.M. Kasim, A. Ali, and S. Shafie. (2012). Unsteady m

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